Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order 2

TL;DR

This paper investigates how limit cycles bifurcate near a double homoclinic loop with a nilpotent saddle of order 2, providing conditions for the number of limit cycles that can appear.

Contribution

It offers new analytical results on the number of limit cycles near complex bifurcation structures involving nilpotent saddles and homoclinic loops.

Findings

Perturbed system can have up to 16 limit cycles near the loop.

Derived conditions for the maximum number of limit cycles.

Validated results with an illustrative example.

Abstract

In this paper, we deal with limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order 2 by studying expansions of the first order Melnikov functions near the loop and coefficients in these expansions. More precisely, we prove that the perturbed system can have 11, 13, 14 or 16 limit cycles in a neighborhood of the loop under certain conditions. Finally, we give an example to illustrate the effectiveness of our main results.